but one hundred years of peace and prosperity is way too fucking long
During the events of New Vegas its actually 92 years.
And they have never even been close to peace and prosperity. Like think about it this way.
In the most prosperous time of the NCRs history(Tandi's reign), shit was still kinda bad. Like, they ended up making an alliance with a powerful crime-family in order to establish greater control over the north, they were conflicting with Vault City, a neighboring city state, and conflict nearly got violent, they were being terrorised by a group of Raiders laying part of there territory under siege, and despite measures being taken to combat corruption, still Cattle Barons held a powerful vote. Plus far-right movements were threatening a dictatorship.
Note that said time was the most prosperous time in there history.
After Tandi's death, they've constantly been in conflict with Tribals, ended up having entire lakes occupied by bandits, lost all efforts to colonise Mexico to Raider Tribes, got themselves in to a brutal war with the Brotherhood of Steel(And after years of fighting one of the most powerful factions on the West Coast didn't even destroy them), They've practically been overtaken by corruption from cattle barons, they are overextended and struggling to hold on to any new territory they absorb.
Let's not mention that they are now in a state of total war against a fascist dictatorship that rivals them as a superpower.
IMHO, they have had enough of there fair of troubles for it to be realistic.
As for it being a shithole, yeah, that's kind of the point.
The world is RUINED.
If you think that's the point of Fallout is that everything is ruined, clearly you missed the point.
The point of Fallout isn't supposed to be that the world is ruined, the point is that it's a new world. When you see The Hub, or Shady Sands, or Junktown, your not supposed to think "Gee whiz, the world sure was ruined", you are supposed to accept that this is the way the world is, and that people are getting on by just by themselves. These places never gave me a feeling of desolation, and ruin, but rather just seemed to be getting by just fine. It wasn't so much pity towards them I felt, so much as acceptance that this is how the world is now.
And lets talk about the endings of Fallout. In the canonical ending, Shady Sands grows in to a republic and spreads across all of California, The Hub maintains its place as a trading giant, The Brotherhood thrive, Junktown becomes a far fairer place, and the gangs end up being cleared up.
That doesn't seem to me to be trying to say "The world will be ruined forever and ever", that seems to be telling me that times are changing, and that the world moves on, places grow and adapt.